Generalized linear mixed model

In statistics, a generalized linear mixed model (GLMM) is an extension to the generalized linear model in which the linear predictor contains random effects in addition to the usual fixed effects.[1][2][3] They also extend the idea of linear mixed models to non-normal data.

Introduction

GLMMs provide a broad range of models for the analysis of grouped data. These models are useful in the analysis of longitudinal data.[4]

Fitting a model

Fitting GLMMs via maximum likelihood (as AIC does) involves integrating over the random effects. In general, those integrals cannot be expressed in analytical form. Various approximate methods have been developed, but none has good properties for all possible models and data sets (e.g. ungrouped binary data are particularly problematic). For this reason, methods involving numerical quadrature or Markov chain Monte Carlo have increased in use, as increasing computing power and advances in methods have made them more practical.

The Akaike information criterion (AIC) is a common criterion for model selection. Estimates of AIC for GLMMs based on certain exponential family distributions have recently been obtained.[5]

Software

See also

References

  1. ↑ Breslow, N. E.; Clayton, D. G. (1993), "Approximate Inference in Generalized Linear Mixed Models", Journal of the American Statistical Association 88 (421): 9–25, doi:10.2307/2290687, JSTOR 2290687
  2. ↑ Stroup, W.W. (2012), Generalized Linear Mixed Models, CRC Press
  3. ↑ Jiang, J. (2007), Linear and Generalized Linear Mixed Models and Their Applications, Springer
  4. ↑ Fitzmaurice, G. M.; Laird, N. M.; Ware, J. H. (2011), Applied Longitudinal Analysis (2nd ed.), John Wiley & Sons, ISBN 0-471-21487-6
  5. ↑ Saefken, B.; Kneib, T.; van Waveren, C.-S.; Greven, S. (2014), "A unifying approach to the estimation of the conditional Akaike information in generalized linear mixed models", Electronic Journal of Statistics 8: 201–225, doi:10.1214/14-EJS881
  6. ↑ Pinheiro, J. C.; Bates, D. M. (2000), Mixed-effects models in S and S-PLUS, Springer, New York
  7. ↑ Berridge, D. M.; Crouchley, R. (2011), Multivariate Generalized Linear Mixed Models Using R, CRC Press